Learn English – What’s the meaning of bee in hand and beauty in eye

This is the first definition given in the Oxford Advanced Learners Dictionary:

1 used in negative sentences and questions, or sentences with if to mean ‘at any time’

Nothing ever happens here.
Don't you ever get tired?
If you're ever in Miami, come and see us.
‘Have you ever thought of changing your job?’ ‘No, never/No I haven't.’
‘Have you ever been to Rome?’ ‘Yes, I have, actually. Not long ago.’
She hardly ever(= almost never) goes out.
We see them very seldom, if ever.
(informal) I'll never ever do that again!

And this is the second

2 used for emphasis when you are comparing things

It was raining harder than ever.
It's my best ever score.

The definition ‘at any time’ in the first definition continues in the second, which is what is in play here—Obama is drawing a comparison (‘more’) between how hard Americans work today and how hard they worked in the past:

So: “Americans are working more than [they worked] at any time [i.e., any time in the past] just to get by.”

On the one hand they'd love to have kids, but on the other, they don't want to give up their freedom.

We do not need to use the phrase on the one hand, before the first sentence if we do not want to. It can make things more interesting for the reader if we do though, because they will be waiting for the new and different piece of information.

However, let's look at the original sentence again:

• On the one hand they'd love to have kids, but on the other, they don't want to give up their freedom.

Notice that because we already used hand in the first part of the sentence we don't need to use it again in the second part. We can just say on the other, instead of on the other hand. But - if we didn't say On the one hand, we need to use the full phrase 'On the other hand', we cannot just say 'on the other':

• They'd love to have kids. *On the other, they don't want to give up their freedom. (wrong)
• They'd love to have kids. On the other hand, they don't want to give up their freedom. (good)

Hope this is helpful!